Question: Solve for $x$ : $ 5|x + 10| - 2 = -1|x + 10| + 9 $
Explanation: Add $ {1|x + 10|} $ to both sides: $ \begin{eqnarray} 5|x + 10| - 2 &=& -1|x + 10| + 9 \\ \\ { + 1|x + 10|} && { + 1|x + 10|} \\ \\ 6|x + 10| - 2 &=& 9 \end{eqnarray} $ Add ${2}$ to both sides: $ \begin{eqnarray} 6|x + 10| - 2 &=& 9 \\ \\ { + 2} &=& { + 2} \\ \\ 6|x + 10| &=& 11 \end{eqnarray} $ Divide both sides by ${6}$ $ \dfrac{6|x + 10|} {{6}} = \dfrac{11} {{6}} $ Simplify: $ |x + 10| = \dfrac{11}{6}$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x + 10 = -\dfrac{11}{6} $ or $ x + 10 = \dfrac{11}{6} $ Solve for the solution where $x + 10$ is negative: $ x + 10 = -\dfrac{11}{6} $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& -\dfrac{11}{6} \\ \\ {- 10} && {- 10} \\ \\ x &=& -\dfrac{11}{6} - 10 \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $6$ $ x = - \dfrac{11}{6} {- \dfrac{60}{6}} $ $ x = -\dfrac{71}{6} $ Then calculate the solution where $x + 10$ is positive: $ x + 10 = \dfrac{11}{6} $ Subtract ${10}$ from both sides: $ \begin{eqnarray} x + 10 &=& \dfrac{11}{6} \\ \\ {- 10} && {- 10} \\ \\ x &=& \dfrac{11}{6} - 10 \end{eqnarray} $ Change the ${ - 10}$ to an equivalent fraction with a denominator of $6$ $ x = \dfrac{11}{6} {- \dfrac{60}{6}} $ $ x = -\dfrac{49}{6} $ Thus, the correct answer is $x = -\dfrac{71}{6} $ or $x = -\dfrac{49}{6} $.